Means

Means


In what follows, x and y are both positive real numbers.

Definition (arithmetic mean). We say that z is the arithmetic mean of x and y if z is one half of the sum of x and y.

Definition (geometric mean). We say that z is the geometric mean of x and y if z is the square root of the product of x and y.

Definition (harmonic mean). We say that z is the harmonic mean of x and y if 1/z is the arithmetic mean of 1/x and 1/y.

These are the three commonest means, which have been known from antiquity. The harmonic mean is always less than or equal to the geometric mean, which in turn is always less than or equal to the arithmetric mean. The mathematical literature contains definitions of many other specific means, and there are also various definitions of abstract means. A fine brief summary of the literature on means is given by Borwein and Borwein (ref. 1 below). Foster and Phillips (in ref. 3 below) give the following definition.

Definition (abstract mean). A mean M is a continuous function of two non-negative real variables such that

  1. min(x,y) is less than or equal to M(x,y), which is less than or equal to max(x,y)

  2. M(x,y) = M(y,x)

  3. if x = M(x,y) then x=y.

The arithmetic, geometric and harmonic means obviously satisfy these properties. So, for example, does the Minkowski mean M, where M(x,y) is the pth root of the arithmetic mean of the pth powers of x and y. Lehmer's means (ref. 4), where the mean of x and y is the sum of the pth powers of x and y divided by the sum of the (p-1)th powers of x and y, also satisfy the above definition. Note that the arithmetic, geometric and harmonic means are all special cases of Lehmer's means, taking p = 1, p=1/2 and p = 0 respectively. But the Chisini means (ref. 2) do not, in general, satisfy the above properties. These are defined as : given a suitable function F, we define M(x,y) = z, where F(z,z) = F(x,y). The arithmetic, geometric and harmonic means are all special cases of the Chisini means.

There are indeed many ways of defining means. One can generate unlimited examples of specific means which satisfy the above definition of an abstract mean by using the following device proposed by Foster and Phillips (ref. 3). Choose any continuous monotonic increasing fuction h. This will therefore have an inverse function. Then, given any mean N belonging to the above class of abstract means, we can construct a "new" mean M belong to this class, by defining

    M(x,y) = z, where h(z) = N(h(x), h(y)).



References

1. J M Borwein and P B Borwein. Pi and the Arithmetic-Geometric Mean , Wiley, 1987.

2. O. Chisini. Sul concetto de media , Period. Mat. 9, 106-116.

3. D M E Foster and G M Phillips. A Generalization of the Archimedean Double Sequence , J. Math. Anal. and Applics. 101, 575-581, 1984.

4. D H Lehmer. On the compounding of certain means , J. Math. Anal. and Applics. 36, 183-200, 1971.